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## What is Maclaurin Series

The Maclaurin series formula is a special case of the Taylor series formula. Or in other words, we can derive the Maclaurin formula from the Taylor series very easily.

Maclaurin series formula helps in writing a function as a series (or sum) of terms involving the derivatives of the function. This formula helps in finding the approximate value of the function.

**Maclaurin series formula**

### e^{x} =1 + x + x^{2}/2! + x^{3}/3! + ……

## How to find Maclaurin Series Formula

**We can find or Derive the Maclaurin series from Taylor series.**

**Sol.** As we Know Taylor series is:

**Let f(x) = e ^{x }and f(0)=1 **(because we only know value of f at x0=1).

**Also f(0) = 1, f’(0) = 1, f’’(0) = 1**

**Since ** **(d/dx)e ^{x} = e^{x}**

**Therefore, f’(x) = e ^{x}**

**F’’(x) = e ^{x}**

**F’’’(x) = e ^{x}**

**:**

**:**

**Now, f(0+h) = f(0) + f’(0)h + f’’(0)h ^{2}/2! + f’’’(0)h^{3}/3!**

**F(h) = 1 + h + h ^{2}/2 + h^{3}/6 + h^{4}/24 + …**

**x+h = 0 +h, ( or replace h by x).**

**f(x) = 1 + x + x ^{2}/2 + x^{3}/6 + x^{4}/24 + ….**

**Or, **This can be simplified as maclaurin series.

*e ^{x} =1 + x + x^{2}/2! + x^{3}/3! + ……*

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